Example: The function f(x) = is continuous on [0, 1]. Is it uniformly continuous there ?

This function (as you could guess from its graph) is uniformly continuous on the closed interval [0, 1]. To prove it, note that

• | f(t) - f(s) | = | t - s | | t + s | < | t - s | 2

because s and t are in the interval [0, 1]. Hence, given any > 0 we can simply choose = / 10 (or something similar) to prove uniform convergence. Can you fill in the details ? A similar argument, incidentally, would work on the interval [0, N] for any number N, but it would fail for the interval [0, ). So, if this function then uniformly continuous on the interval [0, ) ? That's the next example.